Using SubSieve Technique to Accelerate TupleSieve Algorithm

Sieve algorithms are regarded as the best algorithms to solve the shortest vector problem (SVP) on account of its good asymptotical quality, which could make it outperform enumeration algorithms in solving SVP of high dimension. However, due to its large memory requirement, sieve algorithms are not practical as expected, especially on high dimension lattice. To overcome this bottleneck, TupleSieve algorithm was proposed to reduce memory consumption by a trade-off between time and memory. In this work, aiming to make TupleSieve algorithm more practical, we combine TupleSieve algorithm with SubSieve technique and obtain a sub-exponential gain in running time. For 2-tuple sieve, 3-tuple sieve and arbitrary k-tuple sieve, when selecting projection index d appropriately, the time complexity of our algorithm is O(20.415(n-d)), O(20.566(n-d)) and $O(2^{\frac{k\mathrm{log}_2p}{1-k}(n-d)})$ respectively. In practice, we propose a practical variant of our algorithm based on GaussSieve algorithm. Experimental results show that our algorithm implementation is about two order of magnitude faster than FPLLL's GuassSieve algorithm. Moreover, techniques such as XOR-POPCNT trick, progressive sieving and appropriate projection index selection can be exploited to obtain a further acceleration.